Algebraic operations — add, subtract, multiply, divide (rectangular and polar): Visual walkthrough

This page unpacks the boxed result from the parent, the operations note: $z_{1} z_{2} = r_{1} r_{2} [cos (α + β) + i sin (α + β)] .$

Every symbol here is earned below before it is used.

Step 1 — What an "arrow" and its two names are

WHAT. A complex number is a point on flat paper. We draw it as an arrow from the centre (called the origin, the point $0$ ) out to that point.

WHY two names. We can describe the same arrow in two ways, and each way makes a different job easy:

by its shadows: how far right ( $a$ ) and how far up ( $b$ ). We write $z = a + bi$ . The little $i$ is just a flag meaning "this number lives on the up-down axis." Its only rule is $i^{2} = - 1$ .
by its reach and direction: length $r$ (how far the tip is from the origin) and angle $θ$ (how much it is turned, measured anticlockwise from the right-pointing arrow).

PICTURE. The blue arrow below is one complex number shown with both name tags at once.

Figure — Algebraic operations — add, subtract, multiply, divide (rectangular and polar)

The bridge between the two name tags is right-triangle trig: $a = length r fraction that lands sideways cos θ, b = length r fraction that lands upward sin θ .$

Here $cos θ$ is "adjacent over hypotenuse" = the sideways slice of a unit-length turn, and $sin θ$ is "opposite over hypotenuse" = the upward slice. We use them because they are exactly the machine that turns an angle into shadows.

Step 2 — Write both arrows in the "length × direction" form

WHAT. Give the two arrows we want to multiply their polar name tags.

WHY. Multiplication is going to stretch and spin. Stretching acts on the length, spinning acts on the angle — so we must have length and angle written out explicitly before we can watch them move.

PICTURE. Two arrows: a blue one of length $r_{1}$ at angle $α$ , a yellow one of length $r_{2}$ at angle $β$ .

The whole game now is: what is $z_{1} z_{2}$ ?

Step 3 — Multiply the brackets out honestly (FOIL)

WHAT. Just multiply, treating $i$ as an ordinary symbol, using $i^{2} = - 1$ wherever two $i$ 's meet.

WHY. We do not yet know the stretch-and-spin rule — we are going to discover it. So we multiply the raw expressions and see what falls out. No shortcuts.

PICTURE. A "multiplication grid": each term of the first bracket meets each term of the second. Four little products; the one with $i \cdot i$ flips sign.

Pull the two lengths out front (they are just numbers) and expand the unit arrows: $z_{1} z_{2} = r_{1} r_{2} (cos α + i sin α) (cos β + i sin β) .$

The four products in the grid: $real \times real cos α cos β + real \times imag i cos α sin β + imag \times real i sin α cos β + imag \times imag i^{2} sin α sin β .$

The last term carries $i^{2}$ . Replace it by $- 1$ — this is the one place the whole " $i$ magic" enters: $i^{2} sin α sin β = - sin α sin β (a real number now) .$

Step 4 — Sort into real pile and imaginary pile

WHAT. Collect the terms with no $i$ (real) and the terms with one $i$ (imaginary).

WHY. A complex number is one real part + one imaginary part. To read off the answer's shadows we must group them.

PICTURE. The four grid cells sorted into two boxes — a "real" box (two blue cells) and an "imaginary" box (two yellow cells).

$z_{1} z_{2} = r_{1} r_{2} [real part (cos α cos β - sin α sin β) + i imaginary part (sin α cos β + cos α sin β)] .$

The real box got a minus because that term came from $i^{2} = - 1$ .
The imaginary box collected the two "mixed" terms; both share a single factor of $i$ .

Right now this looks like a mess. The next step reveals it is a famous friend in disguise.

Step 5 — Recognise the angle-sum identities (the punchline)

WHAT. Those two boxed expressions are exactly the formulas for the cosine and sine of a combined angle $α + β$ .

WHY these tools. We want the answer's angle. The only expressions that turn " $cos$ and $sin$ of two separate angles" into " $cos$ and $sin$ of one angle" are the angle-sum identities. That is precisely why they show up — the algebra was forced to produce them.

Swap them in: $z_{1} z_{2} = r_{1} r_{2} [cos (α + β) + i sin (α + β)]$

Read the box like a sentence:

$r_{1} r_{2}$ — the new length is the two lengths multiplied → stretch.
$α + β$ — the new angle is the two angles added → spin.

PICTURE. The product arrow (green) sitting at angle $α + β$ , longer than both parents.

Recall Why "add the angles"? — Euler's one-line proof

Using Euler's formula, $cos θ + i sin θ = e^{i θ}$ . Then $z_{1} z_{2} = r_{1} e^{i α} \cdot r_{2} e^{i β} = r_{1} r_{2} e^{i (α + β)}$ . Exponents of a product add — that is the angle addition. The whole Step 3–5 mess is just this hidden.

Step 6 — Where does the angle-sum identity itself come from?

WHAT. We used $cos (α + β)$ and $sin (α + β)$ ; honesty demands we not assume them. Here is the picture-proof.

WHY. The contract: never use a tool before earning it. We earn the identity by rotating a unit arrow.

PICTURE. Start at angle $α$ on the unit circle (point with shadows $cos α, sin α$ ), turn a further $β$ , and drop shadows of the new point $(cos (α + β), sin (α + β))$ . Rotating the shadows of the first point by $β$ reproduces the two identities exactly.

Rotating a point $(cos α, sin α)$ by extra angle $β$ sends its new sideways shadow to $cos α cos β - sin α sin β$ and its new upward shadow to $sin α cos β + cos α sin β$ . Those are the two identities — nothing memorised, just a turn.

Step 7 — Every case: signs, quadrants, and degenerate arrows

WHAT. Confirm the stretch-and-spin rule survives all inputs.

WHY. A rule you only tested in Quadrant I is not proven. Angles can be negative, arrows can be tiny, or one can be zero.

PICTURE. Four mini-panels showing each situation.

The one-picture summary

One frame: two parent arrows (blue length $r_{1}$ angle $α$ , yellow length $r_{2}$ angle $β$ ) and their product (green length $r_{1} r_{2}$ angle $α + β$ ). Lengths multiply, angles add.

Recall Feynman retelling — say it to a 12-year-old

Every complex number is an arrow on graph paper. Each arrow has a length (how far it reaches) and an angle (how much it's turned). When you multiply two arrows, do not add them tip-to-tail — instead play a "stretch and spin" game: the new arrow's length is the two lengths times each other, and its angle is the two angles added up. We didn't just assume this — we multiplied the arrows out letter-by-letter, the little $i$ times itself gave a minus sign, and the leftovers snapped together into the "angle added" formula all on their own. If one arrow has length zero, the answer shrinks to the middle point. If one arrow has length one, nothing stretches — you only spin. That single-arrow spin by a quarter turn is exactly what multiplying by $i$ does.

Connections

Complex plane (Argand diagram) — the paper all these arrows live on.
Modulus and Argument — the length $r$ and angle $θ$ that stretch and spin.
Euler's formula — the one-line reason exponents (angles) add.
De Moivre's Theorem — multiply an arrow by itself $n$ times: length $^{n}$ , angle $\times n$ .
Complex conjugate — flips the angle's sign; the key to the division mirror.
Vectors in 2D — same arrows, but addition there, not multiplication.